Understanding Micropolar Theory in the Earth Sciences I: The Eigenfrequency $$\omega _r$$

Abstract

Even though micropolar theories are widely applied for engineering applications such as the design of metamaterials, applications in the study of the Earth’s interior still remain limited and in particular in seismology. This is due to the lack of understanding of the required elastic material parameters present in the theory as well as the eigenfrequency $$\omega _r$$ ω r which is not observed in seismic data. By showing that the general dynamic equations of the Timoshenko’s beam is a particular case of the micropolar theory we are able to connect micropolar elastic parameters to physically measurable quantities. We then present an alternative micropolar model that, based on the same physical basis as the original model, circumvents the problem of the original eigenfrequency $$\omega _r$$ ω r laking in seismological data. We finally validate our model with a seismic experiment and show it is relevant to explain observed seismic dispersion curves.

Geometric structures of micropolar continuum with elastic and plastic deformations based on generalized Finsler space

“…Then, the curvature tensor and the torsion tensor are obtained from the connection structure, and the geometric approach of the vector bundle can be applied even when the continuum is finitely deformed [28]. Moreover, in generalized continuum such as the Cosserat or micropolar continuum which has often been applied to geophysical field [43–45, 47, 48] and modeling slip, kink, and shear banding [49], the displacement vector is considered as a non-local variable of the line element. In this case, a first-order vector bundle is used to describe the deformation quantities.…”